Let us now discuss the possible fibration structure of \(\mathbb {P}\mathcal {T}\) .
What we will argue is that the family of 2D twistor spaces is fibered over projective spinors if and only if the Einstein
vacuum equations are satisfiedIn twistor theory, the fact that the twistor space of a Ricci-flat complex 4-manifold
is fibered over \(\mathbb {CP}^{1}\)  is already known; see e.g. Remark (2) in p. 444 in [1]}..
Essentially, the obstruction for a fibration is that, given an arbitrary integrable para-Hermitian structure \(K\) , defined
by two projective spinor fields \([\alpha _{A}]\) , \([\beta _{A}]\) , we can achieve
\(\alpha ^{B}\nabla _{BB^{\prime }}\alpha _{A}=0, \qquad \beta ^{B}\nabla _{BB^{\prime }}\beta _{A}=0\) 
